Coaching variables for regression and classification

In a regression or classification setting where we wish to predict Y from x1,x2,..., xp, we suppose that an additional set of ‘coaching’ variables z1,z2,..., zm are available in our training sample. These might be variables that are difficult to measure, and they will not be available when we predict Y from x1,x2,..., xp in the future. We consider two methods of making use of the coaching variables in order to improve the prediction of Y from x1,x2,..., xp. The relative merits of these approaches are discussed and compared in a number of examples.